Constant Reference State Research Articles

Incompressible and fast rotation limit for barotropic Navier–Stokes equations at large Mach numbers

We study the incompressible and fast rotation limit for the barotropic Navier–Stokes equations with Coriolis force, in the case when the Mach number Ma is large with respect to the Rossby number Ro: namely, we focus on the regime Ro≪Ma. For this, we follow a recent approach in Danchin and Mucha (2017) and take also a large bulk viscosity coefficient. We prove that the limit dynamics is described by an incompressible Navier–Stokes type equation, recasted in the vorticity formulation, where however an additional unknown, linked to density oscillations around a fixed constant reference state, comes into play. The proof of the convergence is based on a compensated compactness argument and on the derivation of sharp decay estimates for solutions to a heat equation with fast diffusion in time.

A singularly perturbed nonlinear Poisson–Boltzmann equation: uniform and super‐asymptotic expansions

A steady‐state Poisson–Nernst–Planck system is investigated, which is conformed into a nonlinear Poisson equation by means of the Boltzmann statistics. It describes the electrostatic potential generated by multiple concentrations of ions in a heterogeneous (porous) medium with diluted (solid) particles. The nonlinear elliptic problem is singularly perturbed with the Debye length as a small parameter related to the electric double layer near the solid particle boundary. For star‐shaped solid particles, we prove rigorously that the solution of the problem in spatial dimensions 1d, 2d and 3d is uniformly and super‐asymptotically approximated by a constant reference state. Copyright © 2015 John Wiley & Sons, Ltd.

Consensus with a reference state for fractional-order multi-agent systems

This paper investigates consensus of fractional-order multi-agent systems (MASs) with a reference state. First, a consensus control law with a constant reference state is given using graph theory and stability analysis of fractional-order. Then, a general control law and a particular one for consensus of fractional-order MASs with a time-varying reference state are proposed. Next, the above control laws are extended to solve formation tracking problem. Finally, several simulations are presented to verify the effectiveness of the obtained results.

An inherently mass‐conserving iterative semi‐implicit semi‐Lagrangian discretization of the non‐hydrostatic vertical‐slice equations

AbstractRecently an inherently mass‐conserving semi‐Lagrangian transport scheme has been successfully coupled to an iterative semi‐implicit scheme in a global shallow‐water‐equation (SWE) model. Here that methodology is extended and applied to an iterative semi‐implicit semi‐Lagrangian (SISL) compressible, non‐hydrostatic vertical‐slice model, in which the constant reference state of the SWE model is now replaced by a vertically varying reference state. An advantage of this approach is that it preserves the same basic structure as the interpolating, non‐mass‐conserving, iterative SISL model. The resulting mass‐conserving model is applied to a standard set of test problems for such models and compared with results from both the literature and the interpolating, iterative SISL version of the model. © Crown Copyright 2010. Reproduced with the permission of HMSO. Published by John Wiley & Sons, Ltd.

Multi-vehicle consensus with a time-varying reference state

In this paper, we study the consensus problem in multi-vehicle systems, where the information states of all vehicles approach a time-varying reference state under the condition that only a portion of the vehicles (e.g., the unique team leader) have access to the reference state and the portion of the vehicles might not have a directed path to all of the other vehicles in the team. We first analyze a consensus algorithm with a constant reference state using graph theoretical tools. We then propose consensus algorithms with a time-varying reference state and show necessary and sufficient conditions under which consensus is reached on the time-varying reference state. The time-varying reference state can be an exogenous signal or evolve according to a nonlinear model. These consensus algorithms are also extended to achieve relative state deviations among the vehicles. An application example to multi-vehicle formation control is given as a proof of concept.

Global Existence in Critical Spaces¶for Flows of Compressible¶Viscous and Heat-Conductive Gases

We are concerned with global existence and uniqueness of strong solutions for a general model of viscous and heat-conductive gases. The initial data are supposed to be close to a stable equilibrium with constant density and temperature. Using uniform estimates for the linearized system with a convection term, we get global well-posedness in a functional setting invariant with respect to the scaling of the associated equations (in space dimension N≧3). We also show a smoothing effect on the velocity and the temperature, and a decay on the difference between the density and the constant reference state. These results extend a previous paper devoted to the barotropic case (see [5]).

Global existence in critical spaces for compressible Navier-Stokes equations

We investigate global strong solutions for isentropic compressible fluids with initial data close to a stable equilibrium. We obtain the existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations. More precisely, the initial velocity has the same critical regularity index as for the incompressible homogeneous Navier-Stokes equations, and one more derivative is needed for the density. We point out a smoothing effect on the velocity and a L 1-decay on the difference between the density and the constant reference state. The proof lies on uniform estimates for a mixed hyperbolic/parabolic linear system with a convection term.